Explanation Consider the equation, y ″ + p ( t ) y ′ + q ( t ) y = g ( t ) … ( 1 ) The equation after rewriting, y ″ = − p ( t ) y ′ − q ( t ) y + g ( t ) … ( 2 ) Substitute x 2 for y ′ and x 1 for y in Equation (2). y ″ = − p ( t ) x 2 − q ( t ) x 1 + g ( t ) … ( 3 ) The matrix form of Equation (3) is, [ x 1 x 2 ] ′ = [ 0 1 − q ( t ) − p ( t ) ] [ x 1 x 2 ] + [ 0 g ( t ) ] x ′ = A ( t ) x ( t ) + f ( t ) … ( 4 ) The fundamental matrix for the corresponding homogeneous system is, x ′ = A ( t ) x ( t ) … ( 5 ) The general solution of Equation (5) is, x = X ( t ) c … ( 6 ) Substitute c for u ( t ) in Equation (6). x = X ( t ) u ( t ) … ( 7 ) Differentiate Equation (7). x ′ = X ′ ( t ) u ( t ) + X ( t ) u ′ ( t ) Substitute X ′ ( t ) u ( t ) + X ( t ) u ′ ( t ) for x ′ in Equation (4)

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 9.7, Problem 31E

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The formula for a particular solution yp to the scalar equation y″+p(t)y′+q(t)y=g(t) in terms of two linearly independent solutions y1(t),y2(t) of the corresponding homogeneous equation.

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Chapter 9.7, Problem 30E

Chapter 9.7, Problem 32E

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2.8.1

Do not use the Residue Theorem. Thank you.

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Fundamentals of Differential Equations and Boundary Value Problems

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The formula for a particular solution y p to the scalar equation y ″ + p ( t ) y ′ + q ( t ) y = g ( t ) in terms of two linearly independent solutions y 1 ( t ) , y 2 ( t ) of the corresponding homogeneous equation.

Solution Summary: The author explains the formula for a particular solution y_p to the scalar equation.

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The formula for a particular solution yp to the scalar equation y″+p(t)y′+q(t)y=g(t) in terms of two linearly independent solutions y1(t),y2(t) of the corresponding homogeneous equation.

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